
theorem XCF:
  for c be Complex holds XFS2FS <%c%> = <*c*>
  proof
    let c be Complex;
    A1: len (XFS2FS <%c%>) = len <%c%> by AFINSQ_1:def 9;
    A2: len <%c%> = 1 by AFINSQ_1:def 4; then
    A3: dom (XFS2FS <%c%>) = Seg 1 & dom <*c*> = Seg 1
      by A1,FINSEQ_1:def 3,FINSEQ_1:def 8;
    for k be Nat st k in dom <*c*> holds (XFS2FS <%c%>).k = <*c*>.k
    proof
      let k be Nat;
      assume k in dom <*c*>; then
      k in Seg 1 by FINSEQ_1:def 8; then
      1 <= k <= 1 by FINSEQ_1:1; then
      B3: k = 1 by XXREAL_0:1; then
      <*c*>.k = <%c%>.0
      .= <%c%>.(1-1)
      .= (XFS2FS <%c%>).k by A2,B3,AFINSQ_1:def 9;
      hence thesis;
    end;
    hence thesis by A3;
  end;
